Geometry.Net - Theorems_And_Conjectures: Axiom Of Choice

PSA Presents Axiom Of Choice - 10/11/02
axiom of choice. Performing at Stanford University Friday, October11, 2002 8PM Dinkelspiel Auditorium. More about axiom of choice.
http://psa.stanford.edu/axiom/
Axiom Of Choice
Previous axiomatic set theory Next Axiom of Comprehension. Axiomof Choice. mathematics (AC, or Choice ) An axiom of set theory
http://burks.brighton.ac.uk/burks/foldoc/51/9.htm

Extractions: The Free Online Dictionary of Computing ( http://foldoc.doc.ic.ac.uk/ dbh@doc.ic.ac.uk Previous: axiomatic set theory Next: Axiom of Comprehension mathematics axiom of set theory In other words, we can always choose an element from each set in a set of sets, simultaneously. Function f is a "choice function" for X - for each x in X, it chooses an element of x. Most people's reaction to AC is: "But of course that's true! From each set, just take the element that's biggest, stupidest, closest to the North Pole, or whatever". Indeed, for any finite set of sets, we can simply consider each set in turn and pick an arbitrary element in some such way. We can also construct a choice function for most simple infinite sets of sets if they are generated in some regular way. However, there are some infinite sets for which the construction or specification of such a choice function would never end because we would have to consider an infinite number of separate cases. For example, if we express the real number line R as the union of many "copies" of the rational numbers, Q, namely Q, Q+a, Q+b, and infinitely (in fact uncountably) many more, where a, b, etc. are irrational numbers no two of which differ by a rational, and
TUNES : Axiom Of Choice
axiom of choice. In classical logic, an optional axiom of set theory. Theaxiom of choice is equivalent to Zorn's Lemma or Zermelo's theorem.
http://cliki.tunes.org:8000/Axiom of Choice

Extractions: CLiki pages can be edited by anybody at any time. Imagine a . Double it. Add two. [ Recent Changes ] [ About CLiki ] [ Text Formatting ] [ Create new page ] In classical logic , an optional axiom of set theory. It asserts that out of an arbitrary (even non-enumerable) family (internal set) of non-empty sets, you can find a function that (simultaneously) picks for each set in the family an element of said set. The axiom of choice is equivalent to Zorn's Lemma or Zermelo's theorem. A weaker version often used in classical logic is the Theorem of Compacity. The axiom was long controversial, because it allows reasoning about objects that are non-constructible, and beyond imagination (like unreachable ordinals). It was finally proven that the axiom is independent from the other usual axioms of Set Theory: you can as well admit it or reject it, and end up with as consistent logic models. In a way, it can be argued that classical logic is verily built upon some kind of use of the axiom of choice at the meta-level: simultaneous coherent choice of a truth value for every proposition. Utter rejection of the Axiom of Choice and of the things that it entails has led to the discovery of Intuitionnistic Logic and Constructive Logic, and the development of Proof Theory. A home page for the Axiom of Choice an introduction and links collection by Eric Schechter, Vanderbilt University.
The Axiom Of Choice And Non-enumerable Reals
The axiom of choice and nonenumerable reals. Welcome The writers givetheir version of the axiom of choice 'axiom of choice. Suppose
http://www.angelfire.com/az3/nfold/choice.html

Extractions: [Posted online March 13, 2002; revised July 30, 2002, Aug. 28, 2002, Oct. 12, 2002, Oct. 24, 2002] The following proposition is presented for purposes of discussion. I agree that, according to standard Zermelo-Fraenkel set theory, the proposition is false. Proposition: The Zermelo-Fraenkel power set axiom and the axiom of choice are inconsistent if an extended language is not used to express all the reals. Discussion: The power set axiom reads: 'Given any set X there is a set Y which has as its members all the subsets of X.' The axiom of choice reads: 'For any nonempty set X there is a set Y which has precisely one element in common with set X.'* *Definitions taken from 'Logic for Mathematicians,' A.G. Hamilton, Cambridge, revised 1988. It is known that there is a denumerable set X of writable functions f such that f defines r e R, where R is the nondenumerable set of reals. By writable, we mean f can be written in a language L that has a finite set of operations on a finite set of symbols. In other words, X contains all computable reals.
Infinite Ink: The Axiom Of Choice Quiz
http://ii.best.vwh.net/math/ac/quiz/
Axiom Of Choice
axiom of choice axiom of choice are emigré Persian musicians with their rootsin the repertoire of traditional Persian music. axiom of choice
http://www.hrmusic.com/artists/aocart.html

Extractions: Poignant, innovative, epic, and soulful - these are but a few of the adjectives used to describe the music of Axiom of choice. Formed in 1992 by guitarist, composer, and artistic director Loga Ramin Torkian and co-producer and percussionist Mammad Mohsenzadeh, the ensemble's goal is to define a new sound within the context of Persian music. They were soon joined by vocalist Mamak Khadem whose prodigious vocal talent complemented the group's original compositions. Axiom of Choice... mixes the haunting, warbling vocals of Mamak Khadem with a panoply of Western and Middle Eastern instruments: guitar, electric cello, ney (flute), clarinet, kamancheh (spike violin) duduk and zurna (oboes), saz and divan (strings), and percussion. Torkian also plays a custom-made, seven-string "quarter-tone" guitar with movable frets, which allows him to bridge Western and Eastern scales. Torkian, who studied violin as a child in Iran, then flamenco guitar after immigrating to Eugene, Ore., in 1978 at age 14, writes Axiom's compositions. They are based on traditional Iranian music, updated for a young audience. Western-world fusion guitarists Ralph Towner (and his group, Oregon), and John McLaughlin are major influences. "I played for a while with a traditional ensemble in L.A.," says Torkian, noting that there is significant Iranian population in Southern California. "But it was like taking a lion to the zoo. I felt I had to do something that belonged to the immigrant community, that was a crossover of all my influences, from India, Iran, Japan and Western jazz."
Axiom Of Choice: Contemporary Persian Music
Home Events axiom of choice axiom of choice . . . music that is tender and passionate,floating through the consciousness with the intensity of a dream.
http://www.festivalofsacredmusic.org/festival_2002/events/axiom.html
Re: Axiom Of Choice & QM
Index Re axiom of choice QM. Subject critiqued. Traditionally theaxiom of choice is needed for infinite sets, agreed. However
http://www.lns.cornell.edu/spr/2000-01/msg0020753.html
Re: Axiom Of Choice & QM
Re axiom of choice QM. Subject I've gotten into some tussles with mathematicianson the axiom of choice and physics. Anyway Although
http://www.lns.cornell.edu/spr/2000-01/msg0020712.html

Extractions: Date Prev Date Next Thread Prev Thread Next ... Thread Index Subject From Date : 04 Jan 2000 00:00:00 GMT Approved : spr@rosencrantz.stcloudstate.edu (sci.physics.research) Newsgroups : sci.physics.research Organization : (none) References Follow-Ups From: toby@ugcs.caltech.edu (Toby Bartels) From: Prev by Date: Re: Measurement and Observers Next by Date: Re: gravitational wave detection Prev by thread: ADMINISTRIVIA: Welcome to sci.physics.research! Next by thread: Index(es): Date Thread
Axiom Of Choice
axiom of choice. Schechter, Eric Associate Professor of Mathematics A mathematicianwith major interest in differential equations, real an
http://sources.vanderbilt.edu/head_expert.cfm?HeadingID=4144
Citations: The Consistency Of The Axiom Of Choice And Of The Generalized Continu
K. Godel. The consistency of the axiom of choice and of the generalized continuumhypothesis. Proceedings of the National Academy of Sciences, vol.
http://citeseer.nj.nec.com/context/146735/0

Extractions: This paper is cited in the following contexts: Finitary reductions for local predicativity, I: recursively.. - Tupailo (Correct) ....0 separation, and Ref 9zA z follows from Delta collection. Theorem B [Ar98, Lemma A.2] There is a translation ord from the language L set to the language L 1 such that KP Gamma = T Sigma Gamma ord . Proof (outline) One uses a variant of Godel s hierarchy of constructible sets Let F : Ord L be a surjective mapping given by enumeration of this hierarchy. Using P D axioms, one is able to define Delta formulas b a and a j b in the language L 1 which are equivalent to F (b) 2 F (a) and F (a) F (b) resp. For a formula A 2 L set , A ord denotes the formula ....
Axiom Of Choice In CASL
axiom of choice in CASL. I would vote for explicitly stating in the CASL semantics(and perhaps even in the summary) that we assume the axiom of choice.
http://www.brics.dk/Projects/CoFI/MailingLists/cofi-semantics/2000/msg00004.html

Extractions: Date Prev Date Next Thread Prev Thread Next ... Thread Index http://www.informatik.uni-bremen.de/~till Prev by Date: Paper on relating CASL to other languages Next by Date: Berlin meeting of the semantics task group Prev by thread: Paper on relating CASL to other languages Next by thread: Berlin meeting of the semantics task group Index(es): Date Thread
Axiom Of Choice
axiom of choice. concert report Oct5-2001 San Diego. axiom of choice played aconcert last Friday night at the Neurosciences Institute in San Diego.
http://www.geocities.com/darvaksd/axiom_011005.html

Extractions: Axiom of Choice concert report Oct-5-2001 San Diego Axiom of Choice played a concert last Friday night at the Neurosciences Institute in San Diego. The concert was part of the Mehregan (Autumn festival) celebration, and most of the proceeds went to the victims of the recent tragedies. The opening piece, "Valeh" (poem by Rumi), is a song from Axiom's first album ("Beyond Denial") and started with a duduk solo (in Dashti mode) dedicated to the tragedy victims. This was the first glimpse of the technical mastery and emotional depth of Ruben's playing. As the piece unfolded with Mamak's voice soaring over Loga's guitar arpeggios and Ando's subtle drumming, it was apparent that Axiom is a tight band with well-rehearsed and well-structured songs that nevertheless leave room for spontaneity. "Rhythm Riddle", again from "Beyond Denial", starts in a 10-beat cycle and featured clarinet and voice. Enjoyable. Next Loga played his "quarter-tone guitar" - an instrument that, in his own words, is an ongoing project of his. Using it, he is able to play the modes of Persian music - in this instance, he played Shur. The piece started with a free daramad section, followed by a rhythmic one. Axiom's arrangement of "Baroon, Barooneh", a famous Kurdish folksong with several popular versions, is mellow but rhythmically subtle. Nicely placed between two uptempo songs.
Axiomofchoiceinterview
Cranky Crow World Music. Home Links Profiles axiom of choice Review Axiomof Choice Site. A Conversation with Mamak Khadem, (axiom of choice).
http://www.geocities.com/pherlevi/Axiomofchoiceinterview.htm

Extractions: For many years I have harbored a growing interest in the healing qualities of music. I had read many articles about the healing properties of Mozart and Beethoven and I had read about music from the celestial spheres, but reading expert opinions is one thing and experiencing the healing effects of music even for something as simple as a cold is another story. I sat in a crowded theatre at Meany Hall at the University of Washington campus fighting off chills and certainly not wanting to deal with a crowd, music enthusiasts or not. However, when the members of Axiom of Choice, a group that blends Persian classical music with Western influences hit the stage, I felt my fever abating somewhat as I absorbed the music that emanated from their ancient instruments. The group started the set with Mystic and Fools from their latest release, Unfolding then continued through a set of hypnotic drum beats and flowing, exotic melodies. The audience responded with clapping to the exotic beat on one song and appeared memorized by the performance in general. I spoke with vocalist Mamak Khadem during the group's intricate sound check. Duduk player-clarinetist Ruben Haratoonian's musical gift eddied throughout the green room while Mamak and I discussed the musician's role in achieving world peace and planetary harmony. While it's comforting to know that some people have heard celestial music, many of us have experienced the healing effects of music from various cultures here on earth. And by the way, the next day when I awoke, my cold had left my body and music had replaced it. If music can cure a cold, can it also bring peace to the planet? I believe that it can and so Mamak and I discussed the healing effects of music and the magic of cinema as a way of transcending the chaos of contemporary times.
Axiom Of Choice - Unfolding
Color Of Dreams 10. Messenger Of Time. axiom of choice UnfoldingCD $16.99. Music Box Copyright © 1977- 2003. All rights reserved.
http://www.musicboxla.com/axofchoicun.html
Axiom Of Choice
axiom of choice Unfoldingaxiom of choice - Unfolding. axiom of choice- Niya Yeshaxiom of choice - Niya Yesh. axiom of choice - Axiom
http://www.musicboxla.com/axiomofchoice.html
UW World Series : 2002/2003 Season : Artists : Axiom Of Choice
axiom of choice. axiom of choice is a collective of Persian émigrés and Americancompatriots currently at the forefront of the experimental music world.
http://www.uwworldseries.org/bio-axiom-intro.cfm

Extractions: single tickets: $24 Axiom of Choice is a collective of Persian émigrés and American compatriots currently at the forefront of the experimental music world. With musical roots in the radif (the repertoire of classical Persian music), this innovative ensemble combines exotic instrumentation, soaring vocals and adventurous improvisation in deeply reflective performances that merge East and West, ancient and modern.
The Axiom Of Choice
The axiom of choice. by TJ Jech. bibuq{PSE QA248.J41973}}}, publisher= {\bibnorthhol}, title = {{The axiom of choice}}, year = {1973} }
http://www.itee.uq.edu.au/~bof/Bib/Jech1973a.html
Calculus: The Axiom Of Choice
Before I tell you what the axiom of choice is and how it relates to our modernunderstanding of the world I will first discuss the idea of the axiom in
http://www.wfu.edu/users/escodw0/axiomofchoice.html

Extractions: Well I am glad you decided to come to this page despite its rather intimidating title. I hope that you will find this intriguing and perhaps interesting enough to show off your new found knowledge to your friends. Before I tell you what the Axiom of Choice is and how it relates to our modern understanding of the world I will first discuss the idea of the "axiom" in mathematics. In fact Euclid essentially realized this when he compiled all the knowledge about geometry into his book The Elements . Euclid had five postulates or statements that he could not prove. The first four were obviously unprovable statements like the ability to connect any two points with a line, to draw a circle of any given radius at any point, or the equivalance of any two right angles. Surely these statements just are and there is nothing else that one could ever do to "prove" them to anyones satisfaction short of simply demonstrating to the best of their abilities that a line can be drawn between two point or continued on indefinitely. ON the other hand his fifth postulate sounds like a theorem. It like any other theorem is an if conditional which is clearly based on something more fundamental than its own merit. I will rewrite it in more modern terminology:
Axiom Of Choice - Niya Yesh On Narada Records | International - Department [musi
axiom of choice Niya Yesh on Narada Records cut out cd..melodies and rhythmsof Persia's Radif tradition are combined with Western sensibilities by Southe
http://www.musicaobscura.com/index.cfm/fuseaction/departments/page_number/2/star

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